Basic statistical concepts - confounding

Definition

A third variable that is independently associated with both the exposure and the outcome, and is not on the causal pathway between them
-> creates a spurious or distorted apparent association

1. Associated with the exposure

2. An independent risk factor for the outcome

3. NOT an intermediate step on the causal pathway (if it is, it is a mediator, and adjusting for it removes real effect)

Coffee -> pancreatic cancer, confounded by smoking
Yellow fingers -> lung cancer, confounded by smoking
HRT -> reduced coronary events in observational studies, confounded by healthy user bias - overturned by the WHI randomised trial
Beta-carotene -> less cancer, confounded by diet and lifestyle - randomised trials showed harm

Confounding is a nuisance to be removed; the pooled estimate is biased
Effect modification (interaction) is real biology to be reported; the effect genuinely differs by stratum
- If stratum-specific estimates differ from each other, that is interaction. If they agree with each other but differ from the crude estimate, that is confounding

The single greatest threat to causal inference in observational research
Confounding by indication is the dominant form in clinical epidemiology - sicker patients receive the treatment, so the treatment appears harmful
- The reason observational studies of drugs so often disagree with subsequent trials

Control at the DESIGN stage - preferred

Method	How	Limitation
Randomisation	Balances measured and unmeasured confounders	Only in trials
Restriction	Enrol only non-smokers	Loses generalisability, limits sample
Matching	Pair on the confounder	Cannot then study the matched variable; overmatching risk

Control at the ANALYSIS stage - handles measured confounders only

Method	How
Stratification	Analyse within strata (Mantel-Haenszel pooling)
Multivariable regression	Adjust for covariates simultaneously
Standardisation	Direct or indirect - as in age-standardised mortality rates
Propensity scores	See below
Instrumental variable / Mendelian randomisation	Uses a variable (or a genotype) associated with exposure but not with the outcome except through it - can address unmeasured confounding

Estimate the probability of receiving the treatment or exposure, given a set of observed covariates
Exposed and unexposed individuals with similar propensity scores are then matched, stratified, weighted or adjusted, reducing confounding in non-randomised studies
Advantages: collapses many covariates into one dimension - useful when the outcome is rare but the exposure is common (regression needs ~10 events per covariate)
Fundamental limitation: it can only balance what has been measured. A propensity score analysis is still an observational study and never approaches randomisation

Compare the crude and adjusted estimates. A change of >10% in the effect estimate on adjustment indicates confounding
- Note this is judged on the estimate, not on whether the p value changes
Examine baseline table differences between exposure groups
Ask, for any proposed confounder, the three questions from Description

Were the plausible confounders measured at all? Unmeasured confounding cannot be adjusted away
Were they measured well? (Adjusting for a crudely measured confounder leaves residual confounding - "smoker/non-smoker" does not capture pack-years)
Was anything on the causal pathway adjusted for? (over-adjustment removes true effect)
Was a collider adjusted for? (conditioning on a common effect of exposure and outcome creates bias where none existed - the mechanism behind much "obesity paradox" literature)
Is there a negative control or an E-value quantifying how strong an unmeasured confounder would have to be?

Design out what you can, adjust for what you cannot, and be explicit about what remains
Draw a directed acyclic graph (DAG) before choosing covariates - it distinguishes confounders (adjust), mediators (do not adjust) and colliders (do not adjust)
Pre-specify the adjustment set; choosing covariates by their p value in univariable screening is a discredited practice
Report crude and adjusted estimates side by side
Perform sensitivity analysis - E-value, negative control outcomes, alternative adjustment sets
Where the question is causal and the stakes are high, the answer is a randomised trial - no analytical technique substitutes for it

Bias, effect modification, mediation, collider stratification
Confounding by indication, healthy user and healthy adherer effects, immortal time bias
Propensity score matching and inverse probability weighting
Instrumental variables, Mendelian randomisation
Directed acyclic graphs
Bradford Hill criteria for causation
Simpson paradox - the extreme case, where the direction of the association reverses on stratification

"Adjusted for confounders" is a claim, not a guarantee - always ask which ones, and how well measured
Residual and unmeasured confounding are the default explanation for a surprising observational finding, not the exception
Adjusting for a mediator answers a different question - the direct effect, not the total effect. This is a common and silent error
Propensity score matching discards unmatched patients and changes the population the estimate applies to
The history of preventive medicine is largely a history of confounded observational associations overturned by randomised trials - HRT, beta-carotene, vitamin E, antioxidants

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